The mixed exponential failure process

The mixed exponential failure process is characterized by a time-dependent distribution from which the random hazard rate is obtained directly as an expected value rather than as the ratio of a failure distribution and the corresponding reliability. The properties of the time-dependent distribution are studied. Its mean, variance, and characteristic function (CF) are discussed, and it is shown that the random hazard rate is also expressible in terms of a singularity of the CF in the complex z-plane. Insight into the mixed process is obtained by constructing an electrical filtering analog involving bandpass and low-pass filters. The authors define a nonhomogeneous second-order differential equation for which the mixing distribution is a particular solution. Interdisciplinary applications of the mixing procedure are discussed in terms of photoresponsive detectors which yield generalized Laguerre-polynomial discrete distributions for the photoelectron count